Questions tagged [groupoids]
A groupoid is a small category in which every morphism is an isomorphism. They arise throughout mathematics, e.g. in guise of fundamental groupoids in the theory of covering spaces, holonomy groupoids in the theory of foliations or Lie groupoids in mathematical physics. For groupoids in the sense of universal algebra, i.e., a set with a binary operation, please use the (magma) tag. Also, use other tags such as monoid or category-theory, if needed.
186
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Why do we have $s.\chi_x=\chi_x$ implies that $s(x)=x u$ for some $u \in \mathfrak{C}^{*, 0}$ in an inverse semigroup?
This is a detail in Theorem 4.4 on Page 9 of Xin Li's paper Left Regular Representations of Garside Categories I. C-star-Algebras and Groupoids.
In the proof he said that: $$s.\chi_x=\chi_x \text{ ...
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What categorical notion of "equivalence" does $B:\text{Grp}\to \text{ConnGrpd}$ give?
I've been taking a course on Category Theory recently and really enjoying it. The lecturer today was discussing the notion of equivalence in more detail and showed that every connected groupoid $\...
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∞-Groupoids and the category of elements
Let D(∞-Grpd) be the derived category of infinity groupoids. I'm thinking about the over category and the Grothendieck construction.
D(∞-Grpd)/X ↔ [X, D(∞-Grpd)]
where [C, D] is simply functors. One ...
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$(a^{-1})^{-1} = a$ in a groupoid: proof review
Exercise on groupoids article: Show that in a groupoid, $$
(a^{-1})^{-1} = a \,.
$$
Proof: Every element must have an inverse, and likewise each element’s inverse must itself have an inverse (e.g., $...
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Functors between groupoids
I want to prove that given two groupoids $G$ and $H$, a "functor candidate" $F:G \to H$ between them needs only to preserve compositions, that is, in the case of groupoids, to be a functor ...
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Understanding the "fundamental theorem of covering spaces"
I ask this question because I'm quite confused about the fundamental theorem of covering spaces ;
This theorem say that under suitable hypothesis for a topological space $X$ (semi locally simply ...
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Why is groupoids the appropriate tool for studying quasicrystals?
According to Wikipedia, groupoids is the appropriate tool for studying quasicrystals.
Classical theory of crystals reduces crystals to point lattices where each point is the center of mass of one of ...
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What is a pointed connected groupoid?
Here it says a pointed connected groupoid is a connected groupoid that is pointed. This seems nonsensical: connected groupoids don't have a terminal object, except for the trivial case. The link in ...
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Why, conceptually, isn't the 2-category of connected groupoids equivalent to the 2-category of groups?
It is well-known that every connected groupoid is equivalent to a group. However the 2-category of connected groupoids is not equivalent to the 2-category obtained trivially from the 1-category of ...
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Groupoid embedding into a group [closed]
A necessary condition for a groupoid $(G,∗)$ to embed into a group is that for all $a,b\in G$ then $a ∗ a^{-1}=b∗b^{−1}$.
Question: Is this necessary condition also sufficient?
This question came to ...
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How to prove that a unique inverse exists for every element of this group like structure
Given a quartet $(s,p,m, n)$ and a law
$$(s,p,m,n).(q,r,t,u)=\left(\frac{2}{3}sq, pr,m+(1+t), nu\right)$$
where $s,p,m,n,q,r,t$ and $u$ are in $\mathbb{R} \setminus \{0\}$.
The right and left ...
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Proving a group like object always has a unique right inverse, and a unique left inverse
Given a quartet $(s,p,m, n)$ and a law
$$(s,p,m,n).(q,r,t,u)=(\frac{2}{3}sq, pr,m+(1-t), nu)$$
where $s,p,m,n,q,r,t$ and $u$ are in $\mathbb{R} \setminus \{0\}$.
The left and right identities have ...
- 445
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Name for the group generated by a groupoid
Is there a name for a group generated by a groupoid?
I'm defining a groupoid as a category $\mathcal{C}$ where every arrow has a unique inverse that is both a left inverse and a right inverse.
I'm ...
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Obtaining $\mathscr{B}G$ from the topological groupoid $BG$; which notion of "nerve" of a topological groupoid/category should be used?
For $G$ a group without topology (or a discrete topological group),
let $BG$ denote the groupoid with one object and morphisms given by $G$.
Then, as described at this nLab page,
the geometric ...
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The monoidal structure on the fundamental groupoids of spectrum
I am trying to understand Anderson duality and Picard categories from appendix B of Hopkins and Singer's paper, and I somehow get stuck on Example B.7 (Page 87).
For a spectrum $E$, they consider each ...
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Categorical Intuition of Path Induction
I am trying to understand path induction from the trinitarian point of view. So far I understand the informal intuition of path induction from a homotopical and computational point of view. But I am ...
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Continuity of Norm of Regular Representations on Étale Groupoids
Let $\mathcal G$ be a locally compact, Hausdorff, étale groupoid. Following the notation found in Sims' notes (see here), for $x\in\mathcal G^{(0)}$, let $\lambda_x:C_c(\mathcal G)\to\mathbb B(\ell^2(...
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Is the Mathieu Groupoid $M_{13}$ even special, besides its construction?
The Mathieu groupoid $M_{13}$ has a beautiful construction from a "sliding block puzzle" produced from the projective plane of order $3$; put $12$ labelled counters on $12$ of the $13$ ...
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A groupoid that is not a fundamental groupoid
There is a nice result that says for any group $G$, there exists a topological space $X$ such that the fundamental group $\pi_{1}(X)$ of $X$ is isomorphic to $G$. The proof is not terribly complex: we ...
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Universal cover of a connected groupoid
A group $G$ is a category with exactly one object and all invertible morphisms, which are tue usual elements of the set theoretic group which may be denoted by $|G|$.
Now, for a $G$-set $S$, $\textit{...
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1-dimensional foliation of surfaces with prescribed graph of foliation
Definition of the graph of a foliation
Let we have a $k$ dimensional foliation of an $n$ dimensional manifol $M$. One associates to this foliated manifold a (not necessarily Hausdorff) ...
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Without totality: Semigroupoid, Small Category, and Groupoid - how not be closed possible?
Semigroupoid, Small Category, and Groupoid have group like structures, see https://en.wikipedia.org/wiki/Monoid#Relation_to_category_theory
But there Semigroupoid, Small Category, and Groupoid do not ...
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Groupoid structure of a kernel pair
I would have probably very short and simple question. In https://ncatlab.org/nlab/show/%C4%8Cech+nerve there is a statement
Čech nerve of a morphism $f$ is the internal nerve of the internal groupoid ...
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Functors making functions natural transformations and vice-versa.
I apologize in advance if this is naive.
In this answer Conjugation in a groupoid it is said that given a groupoid $\mathcal G$, and an arbitrary function $\mu:\mathcal G_0\to \bigcup_{x\in \mathcal ...
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(non-)Existence of an endofunctor of $F$ on $g/{\bf B}G^\circlearrowleft$ given the definition of $F$ on objects.
In few words: given an element of a group $g\in G$ given an I'd like to define and endofunctor
$$F:g/{\bf B}G^\circlearrowleft\to g/{\bf B}G^\circlearrowleft$$
I have defined $F$ on objects and I'd ...
- 2,381
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What are groupoids?
As far as I know a groupoid is a category in which every morphism is invertible, however I do not understand this alternative definition:
Let $\mathcal{C}$ be a category with coproducts, $\mathcal{T}$ ...
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What is the practical benefit of a groupoid structure?
Pardon if this question is vague or misguided, I'm a CS person who only dabbles in math.
In a groupoid all morphisms are isomorphisms. So then, any two objects with a morphism between them must be ...
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Is a connected groupoid uniquely determined by its objects and fundamental group?
A small connected groupoid is such a category $G$ that:
$\text{Ob}(G)$ is a nonempty set
$\text{Hom}_G(a,b)$ is a nonempty set for all $a,b\in\text{Ob}(G)$
$f$ is invertible for each $f\in\text{Hom}...
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Classical Galois Theory and Groupoids?
Given your favorite field $k$, we can build its groupoid of algebraic closures in the obvious way: Look at the category of all possible algebraic closures with field isomorphisms as the arrows.
This ...
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Good book for self-study of Magmas/Semigroups/etc.?
I'm currently an undergrad in my second semester of Abstract Algebra. We've covered groups, rings, fields, all that fun stuff. I'm working with Shahriari's "Algebra in Action" as well as ...
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Explicit description of pullback of $(2,1)$-categories.
Let us consider the ordinary category of $(2,1)$-categories. Its objects are groupoid enriched categories, and its morphisms are 2-functors. Is there an explicit way to define the objects, morphisms ...
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Pushout of groupoid
I'm learning category theory. There's a homework question asking for a pushout of groupoid.
Suppose $C_0,C_1,C_2$ are groupoids and denote $f_i:C_0\to C_i\quad i=1,2$ the functor.
I have managed to ...
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Hopf "algebroid" structure of a groupoid convolution algebra?
To male thinks simple as possible, lets say we have a discrete group $G.$ Then the then the group algebra $\mathbb{C}[G]$ (of finitely supported complex valued functions on $G$) has a convolution and ...
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adjunctions from either the category of small categories or the category of groupoids
I'm looking for examples of adjunctions involving either the category Gpd of groupoids or the category Cat of small categories.
The insertion of Gpd into Cat has both a left and a right adjoint, so ...
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Equivalence Between Category of Covering Spaces and Category of Sets with an Action By The Fundamental Groupoid
Let $\Pi_1(X)$ be the fundamental groupoid of a locally path-connected topological space $X$ and define $\Pi_1(X)-\mathbf{Sets}$ to be the category of sets equipped with an action by the fundamental ...
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Functor From Category of Covering Spaces to Category of Sets Equipped With An Action By The Fundamental Groupoid
I have some problems with the understanding of the connection between covering spaces and the fundamental groupoid.
Let $X$ be a topological space and let $\Pi_1(X)$ denote the fundamental groupoid of ...
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Proving that a groupoid is a group, knowing the following properties
(G, ·) is a groupoid. Prove that if it has the following properties it is also a group:
$$1) (a · b) · c = a · (b · c), (\forall)a, b, c \in G;$$
$$2) (∃)u ∈ G : u · a = a · u = a, (∀)a ∈ G;$$
$$3) (∀)...
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The small concrete category $C = (\mathbb{Z_2}, +)$?
I'm grappling with the question of viewing a group as a category, which as I understand it means that if $(G,+)$ is the group in question, then the group can also be thought of as a small concrete ...
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What's the Internal language of a groupoid?
Is there any existing literature on what the internal language of a groupoid might look like?
Please excuse the syntax bashing of this amateur, but i came up with:
symmetry (structural rule):
$$
\frac{...
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classifying space induces a equivalence of categories between PBun$_G(M)$ and $\Pi(M,BG)$ for finite groups $G$
Let $G$ be a finite group, $BG$ its classifying space and M a manifold. Then it is mentioned in https://arxiv.org/abs/1705.05171 (Remark 2.3 d) that there is an equivalence of categories
$$
\Pi (M,BG) ...
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Notation for the scope of definition of a partial binary operation of a groupoid
I have a groupoid ${\displaystyle (G,\ast )}$ with a partial binary operation ${\displaystyle *:G\times G\rightharpoonup G}$.
For every $(a,b)\displaystyle ∈G\times G$, $\displaystyle *$ is defined if ...
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Name for semigroupoid-like structure applicable to a water flow network in graph theory
Hope you are well.
I am studying flow networks (flows of water) in graph theory. By flow network I mean a graph $G = (V,E)$ where $V$ is a set of $ℝ_{>0}$ labelled vertices (water tanks filled with ...
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Group isomorphism between identity morphisms of fundamental groupoid and fundamental group of topological space
For any topological space $X$, I have defined $C=\Pi_1(X)$ by objects $\text{Ob}(C)=X$ and morphisms $\text{Hom}_C(x,y)=\text{HPath}(x,y)$, where $\text{HPath}(x,y)$ denotes the set of homotopy ...
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Homotopy cardinality of weak quotients
Let $G$ be a group (regarded as a one-object groupoid) acting on a groupoid $X$ (i.e.: a functor $F: G \to \mathsf{Groupoids}$ sending the unique object of $G$ to $X$). Denote with $X//G$ the “weak ...
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Definition of Haar systems, especially for groupoids $C^*$-algebra
In A Groupoid approach to $C^*$-algebras Jean Renault introduces left Haar systems. Say $G$ is a groupoid. Let $\Lambda=\{\mu^u,\,\,:\,\,u\in G^0\}$ be a family of Haar measures on $G$. Renault ...
- 489
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Help with understanding the definition of a Hodge groupoid
I am reading this PhD thesis and I can't understand definition 6.21:
(Hodge groupoid). $(X/S)_{Hod}$ is a groupoid whose object object is $X\times \Bbb A^1_S$ and whose morphism object $N_{Hod}$ is ...
- 2,860
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How units of a Lie groupoid act on a manifold?
The definition of a Lie groupoid action given in Sébastien Racanière's notes (up to notation) says that the action of a Lie groupoid $\mathcal{G} \rightrightarrows M$ on a smooth manifold $Q$ consists ...
- 73.8k
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Cancellation law in a Lie groupoid
I'm playing around with the definition of a Lie groupoid following Eckhard Meinrenken's notes. I read the thing for the first time in my life like an hour ago, so assume that I don't know anything ...
- 73.8k
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3
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546
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Simplicial sets which are not Kan complexes
A Kan complex is a simplicial set satisfying the horn-filler condition.
What examples are there of simplicial sets which do not satisfy the horn-filler condition? In particular, what simplicial sets ...
- 3,176
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Every normal subgroupoid is the kernel of a groupoid morphism
Sorry for the block of text from Kirill Mackenzie's chapter on groupoids.
I have proven that $\natural,\natural_0$ is a groupoid morphism and that this quotient construction satisfies the groupoid ...
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